Page 210 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 13: Orthogonality of Solutions

of vectors, i.e. a set of mutually perpendicular vectors. In fact, a function can be considered as a Notes
generalized vector so that fundamental properties of the set of functions are suggested by an
analogous properties of the set of vectors.

Some Basic Definitions

Inner Product: The inner product of two functions f(x) and g(x) is a number defined by the
equation

b
x
g
(f, g) = f ( ) ( ) dx
x
a
on the interval a x b.
Norm of the function: The norm of the function f(x) is defined as the non-negative number

1/2
b
2
f = f ( ) dx
x
a
Orthogonal functions: The condition that the two functions be orthogonal is written as
b
(f, g) = f ( ) ( )dx 0.
x
g
x
a
Orthogonality with respect to a weight (or density) function: The concept of orthogonality can
be extended as follows. Let p(x) 0. Then the condition that the two functions f(x) and g(x) be
orthogonal with respect to the weight function p(x) is written as

b
x
g
x
f
x
p ( ) ( ) ( ) dx = 0
a
Further the norm of the function is defined as
1/2
b
f = p ( ) f 2 ( )dx
x
x
p
a
Again f(x) is said to be normalized when
b
x
x
p ( ) f 2 ( )dx = 1
a
The orthogonality with respect to weight function p(x) can be reduced to the ordinary type by
using the product p ( ) ( ) and p ( ) ( ) as two functions.
x
x
x
x
f
g
Orthogonal Set of Functions:
If we have a set { f (x)}, (n = 1, 2, 3, ...) of real functions defined on an interval a x b, then the
n
{f (x)} is said to be an orthogonal set of functions on the interval a x b if
n
b
x
f ( ) f ( )dx = } 0 when m
x
m n n
a
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